= Solution
For one homogeneous thermoelectric material, the dimensional <thermoelectric figure of merit> is
$$
\boxed{Z=\frac{S^2\sigma}{\kappa}=\frac{S^2}{\rho\kappa}},\qquad[Z]=\mathrm{K^{-1}}.
$$
The dimensionless quantity is $ZT$. The <Seebeck coefficient> $S$ measures the open-circuit voltage response to a temperature difference and reflects the energy asymmetry of charge-carrier transport. The <electrical conductivity> $\sigma$ depends on carrier density and mobility and sets electrical loss; $\rho=1/\sigma$. The <thermal conductivity> is $\kappa=\kappa_e+\kappa_l$, with contributions from carriers and <phonons>, and represents unwanted heat leakage.
Increasing carrier concentration usually raises $\sigma$ while reducing $|S|$ once transport becomes strongly degenerate. The electronic part of $\kappa$ also tends to rise, approximately through the <Wiedemann-Franz law> where applicable. This competition produces an intermediate optimum, commonly in \b[heavily doped <semiconductors>], rather than in a very resistive insulator or an ordinary high-carrier-density metal. The optimum depends on band structure and scattering; the schematic is not a universal measured concentration curve.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2022/paper-1-thermoelectric.png]
{title=Thermoelectric carrier trade-offs, a generator couple and load matching}
{description=Original normalized carrier-property curves, a circuit schematic and the calculated dimensionless load-power curve. The carrier curves illustrate competing trends and do not assert material-specific measurements.}
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